For which closed smooth manifolds does the action of the diffeomorphism group on the set of almost complex structures have exactly one orbit?

For example it is true for $S^2$.


You were lucky to find the only possible example. If you take any manifold of dimension $\ge 4$ you can pick an almost complex structure that is integrable in some closed ball and make it non-integrable outside of it. Since complex $n$-balls have more than one holomorphic structure, we are done. And all surfaces apart from $S^2$ have moduli of complex structures.

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    $\begingroup$ why is there an almost complex structure that is integrable in some closed ball? $\endgroup$
    – user164740
    Oct 7 '20 at 12:16
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    $\begingroup$ Because you can choose an almost complex structure in a ball as you wish, and then extend to the whole manifold. Basically, when you have some almost complex structure on your manifold, you can take a point $p$ and a very small neighbourhood of it with some smooth coordinates. Then in this small neighbourhood $J$ is almost constant. You can perturb it to make it constant $\endgroup$ Oct 7 '20 at 12:28

Dmitri's answer is fine, but there is a different argument that is purely local that is worth bearing in mind as well:

On a $2n$-manifold $M$, the set of almost complex structures on $M$ are the sections of a smooth bundle $\mathscr{J}(M)\to M$ whose fibers are diffeomorphic to $\mathrm{GL}(2n,\mathbb{R})/\mathrm{GL}(n,\mathbb{C})$, a space of real dimension $4n^2 - 2n^2 = 2n^2$.

Thus, almost complex structures in dimension $2n$ depend locally on $2n^2$ functions of $2n$ variables, while diffeomorphisms of $M$ depend locally on $2n$ functions of $2n$ variables. Since $2n^2>2n$ when $n>1$, it follows that, when $n>1$, almost complex structures have local invariants, i.e., the diffeomorphism group cannot act transitively on the space of $k$-jets of almost complex structures for $k$ sufficiently large. Hence, not all almost complex structures can be equivalent under diffeomorphism when $n>1$, even locally.


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